NAME
DATE
4-4
PERIOD
NAME
DATE
4-4
Study Guide and Intervention
Study Guide and Intervention
Graphing Sine and Cosine Functions
|b|
2π
Vertical Shift = d
|b|
Applications of Sinusoidal Functions
Example
The table shows the average monthly
temperatures for Ann Arbor, Michigan, in degrees
Fahrenheit (°F). Write a sinusoidal function that
models the average monthly temperatures as a
function of time x, where x = 1 represents January.
The data can be modeled by a sinusoidal function of the form
y = a sin (bx + c) + d. Find the maximum M and minimum m
values of the data, and use these values to find a, b, c, and d.
1
(M − m)
Amplitude formula
a=−
period
Midline y = d
Example
State the amplitude, period, frequency, phase shift, and vertical
x
π
-−
+ 2. Then graph two periods of the function.
shift of y = -2 cos −
(4
3
)
Amplitude = |a| = |-2| = 2
⎪4⎥
|b|
2π
4
⎪−14 ⎥
y
8π
0 4π 16π 28π 40π x
π
- -−
3
c
4π
= − or −
Phase Shift =- −
3
1
|b|
−
3
( )
3
3
3
Period = 2(xmax - xmin)
Vertical Shift = d or 2
State the amplitude, period, frequency, phase shift, and vertical shift
of each function. Then graph two periods of the function.
π
2
1
π
4
y
1 π
2π 3
π
2. y = cos x - −
+ 2 1, 2π, −, −, 2
(
1. y = 3 sin (2x + π) 3, π, −, - −, 0
4
3
y
)
2
0
−2
π
2
π
3π
2
x
0
−2
4π
3
7π
3
10π x
3
State the frequency and midline of each function.
(2
)
3
2π
1
4π
4π
1
x+−
+ 1 −; y = 1
3. y = 3 sin −
3/16/10 9:36:17 PM
Glencoe Precalculus
Pdf 2nd
−4
−4
3
4. y = cos (3x) - 2 −; y = -2
Write a sine function with the given characteristics.
1
5. amplitude = 2, period = 4, phase shift = −
, vertical shift = 4
( π2
π
4
)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
(
005_042_PCCRMC04_893805.indd 20
Answers
71°
June
80°
July
84°
Aug
81°
Sept
74°
Oct
62°
Vertical shift formula
Nov
48°
M = 84 and m = 30
Dec
35°
2π
⎪b⎥ = −
period
Phase shift formula
π
Phase shift = 4 and ⎪b⎥ = −
6
2π
c = −−
3
Solve for c.
(6
)
π
2π
Therefore, y = 27 sin −
x−−
+ 57 is one model for the average
3
monthly temperature in Ann Arbor, Michigan.
Exercise
1. MUSEUM ATTENDANCE The table gives the number of visitors in
thousands to a museum for each month.
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Visitors
10
8
11
15
24
30
32
29
a. Write a trigonometric function that models the monthly attendance
at the museum using x = 1 to represent January.
(5
10
)
b. According to your model, how many people should the
museum expect to visit during October? 16,292 people
)
20
59°
May
9π
π
y = 12 sin −
x-−
+ 20
Sample answer: y = 1.2 sin x − − + 1
Chapter 4
45°
Apr
6
2
3π
6. amplitude = 1.2, phase shift = −
, vertical shift = 1
3π
2
34°
Mar
xmin = January or month 1
2π
π
⎪b⎥ = −
or −
12
6
c
Phase shift = − −
⎪b⎥
c
4 = −−
π
−
Sample answer: y = 2 sin − x + − + 4
2
30°
Feb
xmax = July or month 7 and
= 2(7 − 1) or 12
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A9
⎪4⎥
Temperature
Jan
M = 84 and m = 30
1
d=−
(M + m)
2
1
= − (84 + 30) or 57
2
1
1
Frequency = − = − or −
2π
2
1
=−
(84 − 30) or 27
2
Month
Answers (Lesson 4-4)
2π
2π
Period = −
=−
or 8π
1
|b|
−
You can use sinusoidal
functions to solve certain application problems.
1
Frequency = − or −
|b|
c
Phase Shift = - −
(continued)
Graphing Sine and Cosine Functions
Transformations of Sine and Cosine Functions A sinusoid is a transformation of
the graph of the sine function. The general form of the sinusoidal functions sine and cosine
are y = a sin (bx + c) + d or y = a cos (bx + c) + d. The graphs of y = a sin (bx + c) + d and
y = a cos (bx + c) + d have the following characteristics.
2π
Amplitude = |a| Period = −
PERIOD
Lesson 4-4
Chapter 4
A01_A19_PCCRMC04_893805.indd 9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
Chapter 4
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005_042_PCCRMC04_893805.indd 21
21
Glencoe Precalculus
10/30/09 6:59:49 PM
DATE
4-4
PERIOD
NAME
DATE
4-4
Practice
Graphing Sine and Cosine Functions
1
g(x) = - −
cos x
The graph of g(x) is the graph of
f(x) compressed vertically. The
1
amplitude of g(x) is −
.
The graph of g(x) is the graph of
f(x) compressed vertically and
reflected in the x-axis. The
1
amplitude of g(x) is −
.
Month Temperature (°F) Month Temperature (°F)
4
Jan
Feb
Mar
Apr
May
June
4
1
y
g (x) = 1 sin x
f (x) = sin x
3
-1
2π
π
x
4π
g (x) = - 1 cos x
y f (x) = cos x
π
-1
3π
4π
x
State the amplitude, period, frequency, phase shift, and vertical shift of each
function. Then graph two periods of the function.
π
3. y = 2 sin x + −
-3
2
1
4. y = −
cos (2x − π) + 2
)
2
A10
2
π
1
frequency = −
;
π ; phase shift = −
2
vertical shift = 2
vertical shift = −3
y
-2π
y
4
4
2
2
π
-π
2π
x
0
-π
-4
-π
2
0
-2
π
2
π
x
-4
Write a sinusoidal function with the given amplitude, period, phase shift, and
vertical shift.
π
5. sine function: amplitude = 15, period = 4π, phase shift = −
, vertical shift = -10
2
(2
)
π
x
y = ±15 sin −
-−
- 10
4
2
y=±−
cos (6x + 2π) + 5
3
7. MUSIC A piano tuner strikes a tuning fork note A above middle C and sets in motion
vibrations that can be modeled by y = 0.001 sin 880tπ. Find the amplitude and period of
the function.
Chapter 4
005_042_PCCRMC04_893805.indd 22
(6
6
5. ROLLER COASTER Part of a roller
coaster track is a sinusoidal function.
The high and low points are separated by
150 feet horizontally and 82 feet
vertically as shown. The low point is
6 feet above the ground.
)
c. According to your model, what is
Baltimore’s average temperature in
July? December?
Sample answer: 77°; 35°
2. BOATING A buoy, bobbing up and down
in the water as waves pass it, moves from
its highest point to its lowest point and
back to its highest point every 10 seconds.
The distance between its highest and
lowest points is 3 feet.
82 ft
6 ft
150 ft
a. Determine the amplitude and period
of a sinusoidal function that models
the bobbing buoy.
a. Write a sinusoidal function that
models the distance the roller coaster
track is above the ground at a given
horizontal distance x.
1.5; 10 s
b. Write an equation of a sinusoidal
function that models the bobbing buoy,
using x = 0 as its highest point.
π
y = 41 cos −
x + 47
150
b. Point A is 40 feet to the right of the
y-axis. How far above the ground is
the track at point A? 74.4 ft
(5 )
1
0.001; −
440
22
2
Sample answer:
5π
π
x+−
+ 54.5
y = 22.5 cos −
Sample answer:
π
y = 1.5 cos −
x
Glencoe Precalculus
Chapter 4
10/30/09 6:27:27 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
π
π
2
6. cosine function: amplitude = −
, period = −
, phase shift = - −
, vertical shift = 5
3
3
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
frequency = −
;
2π π
phase shift = − −;
2
4. SWING Marsha is pushing her brother
Bobby on a rope swing over a creek.
When she starts the swing, he is 7 feet
over land away from the edge of the
creek. After 2 seconds, Bobby is 11 feet
over the water past the edge of the creek.
Assume that the distance from the edge of
the creek varies sinusoidally with time
and that the distance y is positive when
Bobby is over the water and negative when
he is over land. Write a trigonometric
function that models the distance
Bobby is from the edge of the creek at
π
time t seconds. y = -9 cos − t + 2
b. Write an equation of a sinusoidal
function that models the monthly
temperatures.
1
amplitude = −
; period = π;
amplitude = 2; period = 2π;
Sample answer: c is a positive
multiple of n.
77
76
69
57
47
37
amplitude = 22.5; period = 12
months; phase shift: -5
months; vertical shift = 54.5°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(
July
Aug
Sept
Oct
Nov
Dec
a. Determine the amplitude, period,
phase shift, and vertical shift of a
sinusoidal function that models the
monthly temperatures using x = 1 to
represent January.
4
2π
32
35
44
53
63
73
005_042_PCCRMC04_893805.indd 23
Answers (Lesson 4-4)
0
1
0
3. A student graphed a periodic function
with a period of n. The student then
translated the graph c units to the right
and obtained the original graph. Describe
the relationship between c and n.
1. METEOROLOGY The average monthly
temperatures for Baltimore, Maryland,
are shown below.
2. f(x) = cos x
1
g(x) = −
sin x
3
3
Word Problem Practice
Graphing Sine and Cosine Functions
Describe how the graphs of f(x) and g(x) are related. Then find the amplitude
of g(x) and sketch two periods of both functions on the same coordinate axes.
1. f(x) = sin x
PERIOD
Lesson 4-4
Chapter 4
NAME
23
Glencoe Precalculus
10/28/09 10:37:52 PM